Phase Portraits of Hyperbolic Geometry

Phase Portraits of Linear Systems

1.1. One positive and one negative eigenvalue. This is described in Example 4.2.1 of [1]. Here the origin is a saddle: unstable but neither a repeller nor an attractor. Draw the two eigenlines (the lines defined by eigenvectors). Put two “outward” arrows on the eigenline that corresponds to the positive eigenvalue and two “inward” arrows on the eigenline for the negative eigenvalue. Fill in the...

Hyperbolic Geometry

Global phase portraits of a SIS model

In the qualitative theory of ordinary differential equations, we can find many papers whose objective is the classification of all the possible topological phase portraits of a given family of differential system. Most of the studies rely on systems with real parameters and the study consists of outlining their phase portraits by finding out some conditions on the parameters. Here, we studied a...

Phase portraits of uniform isochronous quartic centers

In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...